Question: Is ${745834}$ divisible by $9$ ?
Explanation: A number is divisible by $9$ if the sum of its digits is divisible by $9$ . [ Why? First, we can break the number up by place value: $ \begin{eqnarray} {745834}= &&{7}\cdot100000+ \\&&{4}\cdot10000+ \\&&{5}\cdot1000+ \\&&{8}\cdot100+ \\&&{3}\cdot10+ \\&&{4}\cdot1 \end{eqnarray} $ Next, we can rewrite each of the place values as $1$ plus a bunch of $9$ s: $ \begin{eqnarray} {745834}= &&{7}(99999+1)+ \\&&{4}(9999+1)+ \\&&{5}(999+1)+ \\&&{8}(99+1)+ \\&&{3}(9+1)+ \\&&{4} \end{eqnarray} $ Now if we distribute and rearrange, we get this: $ \begin{eqnarray} {745834}= &&\gray{7\cdot99999}+ \\&&\gray{4\cdot9999}+ \\&&\gray{5\cdot999}+ \\&&\gray{8\cdot99}+ \\&&\gray{3\cdot9}+ \\&& {7}+{4}+{5}+{8}+{3}+{4} \end{eqnarray} $ Any number consisting only of $9$ s is a multiple of $9$ , so the first five terms must all be multiples of $9$ That means that to figure out whether the original number is divisible by $9 $ , all we need to do is add up the digits and see if the sum is divisible by $9$ . In other words, ${745834}$ is divisible by $9$ if ${ 7}+{4}+{5}+{8}+{3}+{4}$ is divisible by $9$ Add the digits of ${745834}$ $ {7}+{4}+{5}+{8}+{3}+{4} = {31} $ If ${31}$ is divisible by $9$ , then ${745834}$ must also be divisible by $9$ ${31}$ is not divisible by $9$, therefore ${745834}$ must not be divisible by $9$.